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Re: Mathematica as a New Approach to Teaching Maths

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  • Subject: [mg127465] Re: Mathematica as a New Approach to Teaching Maths
  • From: "djmpark" <djmpark at comcast.net>
  • Date: Fri, 27 Jul 2012 04:56:35 -0400 (EDT)
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Thanks for mentioning the Conrad Wolfram Ted Talk.

http://www.ted.com/talks/lang/en/conrad_wolfram_teaching_kids_real_math_with
_computers.html 

I think Conrad's talk was correct in the main points, glib about the
difficulties, correct about the potential.

1) The first point is that Mathematica is difficult. There is a long
learning curve. Therefore students headed for careers with technical content
should start learning Mathematica early, long before they have to tackle
difficult mathematical material. The students would learn some mathematics
as they are learning Mathematica but it is a question as to what subjects
should be treated when the main objective is to learn Mathematica. (If the
aim is to learn real mathematics John Stillwell's "Numbers and Geometry"
might be a good source of material.)

2) The test of having skill with Mathematica is the ability to turn real
world problems into Mathematica specifications - one of Conrad's points. I
call it flying solo, as opposed to copying someone else's code.

3) Conrad disparages learning math by hand with paper and pencil. I would
rather co-opt the paradigm. Don't think of Mathematica as a super graphical
calculator, or as a programming language (although it is in part these
things) but think of it as a piece of paper on which you are writing your
ideas, exploring them and presenting them. It is indeed a magic piece of
paper with its active calculation, memory and dynamics - but still a piece
of paper.

4) This means that students should also learn how to use the Sectional
structure of notebook and discuss their material and its development in Text
cells. It means that material will often have to be presented in stages with
multiple definitions and derivations, graphical presentations and dynamic
presentations. One cannot often present coherent material in a single
Manipulate statement with lots of Sliders.

5) I didn't quite follow the example that Conrad presented in place of a
test. I'm not certain if the student was just to use the dynamic
presentation or if  he was to design and implement it. The first might not
be especially instructive and the second might be too difficult. In any
case, I would say that the tests or homework should be in the form of essay
questions using the techniques of the preceding point. If a student writes
an essay notebook he has something to keep, refer to in the future, and show
off.

6) The analogy of jumping over a chasm is one I presented on MathGroup in
2008. http://forums.wolfram.com/mathgroup/archive/2008/Nov/msg00714.html 

7) In general, the entire topic of using Mathematica in education is quite
difficult because the capabilities that Mathematica brings are so
revolutionary. It is all too easy to be unconsciously mired in old paradigms
or to fall into the pit of "computer junk". In some cases full-fledged
Applications will be necessary, done in Workbench with documentation and
examples or course material. Students must learn how to write routines (and
their usage messages), but maybe not every extra routine convenient for some
subject matter. Such applications should not put the student into a box but
rather provide a set of routines that supplement and extend regular
Mathematica. An example might be a set of axioms. in the form of rules or
routines that apply the rules, for some field of mathematics. Then a student
could do derivations or proofs using the axioms. What better way to become
familiar with them using them and seeing them in action.

8) Some things along these lines are in the Presentations Application. One
of the things students have most difficulty with is custom graphics because
the WRI paradigm is really convoluted when it comes to combining things or
making geometrical diagrams. Presentations tries to fix that. There is a
section on single variable integrals that allows a student to do various
manipulations on the integrals such as change of variable, integration by
parts or trigonometric substitution so they can see what is happening. There
is a Students Linear Equations section that allows matrices to be
manipulated with primitive commands and see the results. The matrices also
have row and column labels to give them context. I've been working with John
Browne's GrassmannAlgebra Application, primarily trying to learn it but also
helping with interface and the writing of some introductory examples. This
Application would be great for teaching plane geometry because one can
easily define points, lines, triangles and other objects algebraically; do
things such as calculate lengths, areas and angles; rotate and translate
objects; calculate perpendiculars and find intersections; or determine if a
point is inside or outside a triangle - all with algebra. One can also draw
the geometric diagrams directly using the Grassmann algebra expressions for
the coordinates.

It's the kind of thing that can be done but it's more than regular
Mathematica and it takes development.


David Park
djmpark at comcast.net 
http://home.comcast.net/~djmpark/index.html 





From: amzoti [mailto:amzoti at gmail.com] 


Hi All,

I just watched what is probably considered a hot button topic issue by some
from "Conrad Wolfram's recent TED talk "Stop teaching calculating, start
teaching math".

I was wondering if any Mathematica users have ever explored this and how
they may be approaching it.

I love the idea of teaching students to use Mathematica as an exploratory
tool which allows them to ask what if questions for learning to problem
solve and to ask better questions.

Has anyone developed or researched an approach to replace the traditional
teaching methods (crank out silly answers) at any level?

It would be great if Mathematica could even suggest such as approach!

Anyway, would love to hear any feedback, pointers or ideas.

Sorry if this is off-topic!

Thanks




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